Rise Over Run Formula: Calculating Linear Slope, Incline Angles, Roof Pitch, and Percent Grade
The rise over run formula calculates linear slope as $m = \text{Rise} / \text{Run} = (y_2 - y_1) / (x_2 - x_1)$. Rise represents the vertical displacement (Δy), and Run represents the horizontal displacement (Δx). To find the angle of inclination, evaluate $\theta = \arctan(m) \times (180 / \pi)$. Grade percentage equals $m \times 100\%$, and construction roof pitch expresses vertical rise per 12 inches of run ($m \times 12$).
Key Takeaway Facts
- Rise (vertical change) and Run (horizontal change) must always be measured in identical units before dividing.
- A completely horizontal line has a slope of zero (m = 0, angle = 0°).
- A completely vertical line has an undefined slope because dividing by a horizontal run of zero is impossible.
- Under the Americans with Disabilities Act (ADA), public wheelchair ramps must not exceed a 1:12 slope (8.33% grade).
- In US residential carpentry, roof pitch is standardized as the vertical rise per 12 inches of horizontal run (e.g., 6:12).
Rise Over Run Calculator
Calculate slope, angle of incline, grade %, and roof pitch.
1. Mathematical Definitions: From Cartesian Coordinates to Real-World Slopes
Frequently Asked Questions
What is a 6 in 12 roof pitch in degrees?
A 6:12 roof pitch has a decimal slope of 6/12 = 0.5000. In degrees, θ = arctan(0.5000) ≈ 26.57 degrees, representing a 50% grade.
Editorial Review & Fact-Checking Assurance
This guide was researched and drafted by the ToolQix Engineering & Applied Geometry Group and technically verified by Licensed Professional Civil Engineer (PE) under ToolQix's strict accuracy protocols. Formulas, calculations, and instructions were independently tested against current industry specifications.